Optimal. Leaf size=103 \[ -\frac{(b d-a e) (-3 a B e+2 A b e+b B d)}{b^4 (a+b x)}-\frac{(A b-a B) (b d-a e)^2}{2 b^4 (a+b x)^2}+\frac{e \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^4}+\frac{B e^2 x}{b^3} \]
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Rubi [A] time = 0.216599, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{(b d-a e) (-3 a B e+2 A b e+b B d)}{b^4 (a+b x)}-\frac{(A b-a B) (b d-a e)^2}{2 b^4 (a+b x)^2}+\frac{e \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^4}+\frac{B e^2 x}{b^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^2)/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{e^{2} \int B\, dx}{b^{3}} + \frac{e \left (A b e - 3 B a e + 2 B b d\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{\left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right )}{b^{4} \left (a + b x\right )} - \frac{\left (A b - B a\right ) \left (a e - b d\right )^{2}}{2 b^{4} \left (a + b x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**2/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.125777, size = 140, normalized size = 1.36 \[ \frac{B \left (-5 a^3 e^2+2 a^2 b e (3 d-2 e x)+a b^2 \left (-d^2+8 d e x+4 e^2 x^2\right )+2 b^3 x \left (e^2 x^2-d^2\right )\right )+2 e (a+b x)^2 \log (a+b x) (-3 a B e+A b e+2 b B d)-A b (b d-a e) (3 a e+b (d+4 e x))}{2 b^4 (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x)^3,x]
[Out]
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Maple [B] time = 0.013, size = 242, normalized size = 2.4 \[{\frac{B{e}^{2}x}{{b}^{3}}}+{\frac{{e}^{2}\ln \left ( bx+a \right ) A}{{b}^{3}}}-3\,{\frac{{e}^{2}\ln \left ( bx+a \right ) Ba}{{b}^{4}}}+2\,{\frac{e\ln \left ( bx+a \right ) Bd}{{b}^{3}}}+2\,{\frac{aA{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{Ade}{{b}^{2} \left ( bx+a \right ) }}-3\,{\frac{B{a}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}+4\,{\frac{Bade}{{b}^{3} \left ( bx+a \right ) }}-{\frac{B{d}^{2}}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{a}^{2}A{e}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{aAde}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{A{d}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{B{a}^{3}{e}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{2}de}{{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{d}^{2}}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^2/(b*x+a)^3,x)
[Out]
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Maxima [A] time = 1.34876, size = 230, normalized size = 2.23 \[ \frac{B e^{2} x}{b^{3}} - \frac{{\left (B a b^{2} + A b^{3}\right )} d^{2} - 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} d e +{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} + 2 \,{\left (B b^{3} d^{2} - 2 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d e +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{2}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{{\left (2 \, B b d e -{\left (3 \, B a - A b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220396, size = 348, normalized size = 3.38 \[ \frac{2 \, B b^{3} e^{2} x^{3} + 4 \, B a b^{2} e^{2} x^{2} -{\left (B a b^{2} + A b^{3}\right )} d^{2} + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} d e -{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{2} - 2 \,{\left (B b^{3} d^{2} - 2 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d e + 2 \,{\left (B a^{2} b - A a b^{2}\right )} e^{2}\right )} x + 2 \,{\left (2 \, B a^{2} b d e -{\left (3 \, B a^{3} - A a^{2} b\right )} e^{2} +{\left (2 \, B b^{3} d e -{\left (3 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (2 \, B a b^{2} d e -{\left (3 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.9451, size = 187, normalized size = 1.82 \[ \frac{B e^{2} x}{b^{3}} - \frac{- 3 A a^{2} b e^{2} + 2 A a b^{2} d e + A b^{3} d^{2} + 5 B a^{3} e^{2} - 6 B a^{2} b d e + B a b^{2} d^{2} + x \left (- 4 A a b^{2} e^{2} + 4 A b^{3} d e + 6 B a^{2} b e^{2} - 8 B a b^{2} d e + 2 B b^{3} d^{2}\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac{e \left (- A b e + 3 B a e - 2 B b d\right ) \log{\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**2/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.223041, size = 209, normalized size = 2.03 \[ \frac{B x e^{2}}{b^{3}} + \frac{{\left (2 \, B b d e - 3 \, B a e^{2} + A b e^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{B a b^{2} d^{2} + A b^{3} d^{2} - 6 \, B a^{2} b d e + 2 \, A a b^{2} d e + 5 \, B a^{3} e^{2} - 3 \, A a^{2} b e^{2} + 2 \,{\left (B b^{3} d^{2} - 4 \, B a b^{2} d e + 2 \, A b^{3} d e + 3 \, B a^{2} b e^{2} - 2 \, A a b^{2} e^{2}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^2/(b*x + a)^3,x, algorithm="giac")
[Out]